Midterms. PAC Learning SVM Kernels+Boost Decision Trees. MultiClass CS446 Spring 17

Transcription

1 Midterms PAC Learning SVM Kernels+Boost Decision Trees 1

2 Grades are on a curve Midterms Will be available at the TA sessions this week Projects feedback has been sent. Recall that this is 25% of your grade! 2

3 Classification So far we focused on Binary Classification For linear models: Perceptron, Winnow, SVM, GD, SGD The prediction is simple: Given an example x, Prediction = sgn(w T x) Where w is the learned model The output is a single bit 3

4 Multi-Categorical Output Tasks Multi-class Classification (y {1,...,K}) character recognition ( 6 ) document classification ( homepage ) Multi-label Classification (y {1,...,K}) document classification ( (homepage,facultypage) ) Category Ranking (y (K)) user preference ( (love > like > hate) ) document classification ( hompage > facultypage > sports ) Hierarchical Classification (y {1,..,K}) cohere with class hierarchy place document into index where soccer is-a sport 4

5 Setting Learning: Given a data set D = {(x i, y i )} m 1 Where x i 2 R n, y i 2 {1,2,,k}. Prediction (inference): Given an example x, and a learned function (model), Output a single class labels y. 5

6 Binary to Multiclass Most schemes for multiclass classification work by reducing the problem to that of binary classification. There are multiple ways to decompose the multiclass prediction into multiple binary decisions One-vs-all All-vs-all Error correcting codes We will then talk about a more general scheme: Constraint Classification It can be used to model other non-binary classification schemes and leads to Structured Prediction. 6

7 One-Vs-All Assumption: Each class can be separated from all the rest using a binary classifier in the hypothesis space. Learning: Decomposed to learning k independent binary classifiers, one for each class label. Learning: Let D be the set of training examples. 8 label l, construct a binary classification problem as follows: Positive examples: Elements of D with label l Negative examples: All other elements of D This is a binary learning problem that we can solve, producing k binary classifiers w 1, w 2, w k Decision: Winner Takes All (WTA): f(x) = argmax i w it x 7

8 Solving MultiClass with 1vs All learning MultiClass classifier Function f : R n {1,2,3,...,k} Decompose into binary problems Not always possible to learn No theoretical justification Need to make sure the range of all classifiers is the same (unless the problem is easy) MultiClass CS446 Spring 17 8

9 Learning via One-Versus-All (OvA) Assumption Find v r,v b,v g,v y R n such that v r.x > 0 iff y = red v b.x > 0 iff y = blue v g.x > 0 iff y = green v y.x > 0 iff y = yellow Classification: f(x) = argmax i v i x H = R nk Real Problem MultiClass CS446 Spring 17 9

10 All-Vs-All Assumption: There is a separation between every pair of classes using a binary classifier in the hypothesis space. Learning: Decomposed to learning [k choose 2] ~ k 2 independent binary classifiers, one corresponding to each pair of class labels. For the pair (i, j): Positive example: all exampels with label i Negative examples: all examples with label j Decision: More involved, since output of binary classifier may not cohere. Each label gets k-1 votes. Decision Options: Majority: classify example x to take label i if i wins on x more often than j (j=1, k) A tournament: start with n/2 pairs; continue with winners. 10

11 Learning via All-Verses-All (AvA) Assumption Find v rb,v rg,v ry,v bg,v by,v gy R d such that v rb.x > 0 if y = red < 0 if y = blue v rg.x > 0 if y = red < 0 if y = green... (for all pairs) It is possible to separate all k classes with the O(k 2 ) classifiers H = R kkn How to classify? Individual Classifiers Decision Regions MultiClass CS446 Spring 17 11

12 Classifying with AvA Tournament Majority Vote 1 red, 2 yellow, 2 green? All are post-learning and might cause weird stuff MultiClass CS446 Spring 17 12

13 One-vs-All vs. All vs. All Assume m examples, k class labels. For simplicity, say, m/k in each. One vs. All: classifier f i : m/k (+) and (k-1)m/k (-) Decision: Evaluate k linear classifiers and do Winner Takes All (WTA): All vs. All: f(x) = argmax i f i (x) = argmax i w it x Classifier f ij : m/k (+) and m/k (-) More expressivity, but less examples to learn from. Decision: Evaluate k 2 linear classifiers; decision sometimes unstable. What type of learning methods would prefer All vs. All (efficiency-wise)? (Think about Dual/Primal) MultiClass CS446 Spring 17 13

14 Error Correcting Codes Decomposition 1-vs-all uses k classifiers for k labels; can you use only log 2 k? Reduce the multi-class classification to random binary problems. Choose a code word for each label. K=8: all we need is 3 bits, three classifiers Rows: An encoding of each class (k rows) Columns: L dichotomies of the data, each corresponds to a new classification problem Extreme cases: Label P1 P2 P3 P4 1-vs-all: k rows, k columns k rows log 2 k columns Each training example is mapped to one example per column (x,3) {(x,p1), +; (x,p2), -; (x,p3), -; (x,p4), +} To classify a new example x: Evaluate hypothesis on the 4 binary problems {(x,p1), (x,p2), (x,p3), (x,p4),} Choose label that is most consistent with the results. Use Hamming distance (bit-wise distance) Nice theoretical results as a function of the performance of the P i s (depending on code & size) Potential Problems? Can you separate any dichotomy? k

15 Problems with Decompositions Learning optimizes over local metrics Does not guarantee good global performance We don t care about the performance of the local classifiers Poor decomposition poor performance Difficult local problems Irrelevant local problems Especially true for Error Correcting Output Codes Another (class of) decomposition Difficulty: how to make sure that the resulting problems are separable. Efficiency: e.g., All vs. All vs. One vs. All Former has advantage when working with the dual space. Not clear how to generalize multi-class to problems with a very large # of output variables. 15

16 1 Vs All: Learning Architecture k label nodes; n input features, nk weights. Evaluation: Winner Take All Training: Each set of n weights, corresponding to the i-th label, is trained Independently, given its performance on example x, and Independently of the performance of label j on x. Hence: Local learning; only the final decision is global, (Winner Takes All (WTA)). However, this architecture allows multiple learning algorithms; e.g., see the implementation in the SNoW/LbJava Multi-class Classifier Targets (each an LTU) Weighted edges (weight vectors) Features MultiClass CS446 Spring 17 21

17 Recall: Winnow s Extensions Winnow learns monotone Boolean functions We extended to general Boolean functions via Balanced Winnow 2 weights per variable; Decision: using the effective weight, the difference between w + and w - This is equivalent to the Winner take all decision Positive w + Learning: In principle, it is possible to use the 1-vs-all rule and update each set of n weights separately, but we suggested the balanced Update rule that takes into account how both sets of n weights predict on example x If [(w w ) x ] y, w i w i r y x i, w i w i r y x i Negative w - Can this be generalized to the case of k labels, k >2? We need a global learning approach 22

18 Extending Balanced In a 1-vs-all training you have a target node that represents positive examples and target node that represents negative examples. Typically, we train each node separately (mine/not-mine example). Rather, given an example we could say: this is more a + example than a example. If [(w w ) x ] y, w i w i r y x i, w i w i r y x i We compared the activation of the different target nodes (classifiers) on a given example. (This example is more class + than class -) Can this be generalized to the case of k labels, k >2? 23

19 Where are we? Introduction Combining binary classifiers One-vs-all All-vs-all Error correcting codes Training a single (global) classifier Multiclass SVM Constraint classification 24

20 Recall: Margin for binary classifiers The margin of a hyperplane for a dataset is the distance between the hyperplane and the data point nearest to it Margin with respect to this hyperplane 25

21 Multiclass Margin 26

22 Multiclass SVM (Intuition) Recall: Binary SVM Maximize margin Equivalently, Minimize norm of weight vector, while keeping the closest points to the hyperplane with a score 1 Multiclass SVM Each label has a different weight vector (like one-vs-all) Maximize multiclass margin Equivalently, Minimize total norm of the weight vectors while making sure that the true label scores at least 1 more than the second best one. 27

23 Multiclass SVM in the separable case Recall hard binary SVM Size of the weights. Effectively, regularizer The score for the true label is higher than the score for any other label by 1 28

24 Multiclass SVM: General case Size of the weights. Effectively, regularizer Total slack. Effectively, don t allow too many examples to violate the margin constraint The score for the true label is higher than the score for any other label by 1 Slack variables. Not all examples need to satisfy the margin constraint. Slack variables can only be positive 29

25 Multiclass SVM: General case Size of the weights. Effectively, regularizer Total slack. Effectively, don t allow too many examples to violate the margin constraint The score for the true label is higher than the score for any other label by 1 -» i Slack variables. Not all examples need to satisfy the margin constraint. Slack variables can only be positive 30

26 Multiclass SVM Generalizes binary SVM algorithm If we have only two classes, this reduces to the binary (up to scale) Comes with similar generalization guarantees as the binary SVM Can be trained using different optimization methods Stochastic sub-gradient descent can be generalized Try as exercise 31

27 Multiclass SVM: Summary Training: Optimize the global SVM objective Prediction: Winner takes all argmax i w it x With K labels and inputs in < n, we have nk weights in all Same as one-vs-all Why does it work? Why is this the right definition of multiclass margin? A theoretical justification, along with extensions to other algorithms beyond SVM is given by Constraint Classification Applies also to multi-label problems, ranking problems, etc. [Dav Zimak; with D. Roth and S. Har-Peled] 32

28 Constraint Classification The examples we give the learner are pairs (x,y), y 2 {1, k} The black box learner (1 vs. all) we described might be thought of as a function of x only but, actually, we made use of the labels y How is y being used? y decides what to do with the example x; that is, which of the k classifiers should take the example as a positive example (making it a negative to all the others). How do we predict? Is it better in any well defined way? Let: f y (x) = w yt x Then, we predict using: y * = argmax y=1, k f y (x) Equivalently, we can say that we predict as follows: Predict y iff 8 y 2 {1, k}, y :=y (w yt w y T ) x 0 (**) So far, we did not say how we learn the k weight vectors w y (y = 1, k) Can we train in a way that better fits the way we predict? What does it mean? 33

29 We showed: if pairs of labels are separable (a reasonable assumption) than in some higher dimensional space, the problem is linearly separable. Linear Separability for Multiclass We are learning k n-dimensional weight vectors, so we can concatenate the k weight vectors into Notice: This is just a representational w= (w 1, w 2, w k ) 2 R nk trick. We did not say how to learn the weight vectors. Key Construction: (Kesler Construction; Zimak s Constraint Classification) We will represent each example (x,y), as an nk-dimensional vector, x y, with x embedded in the y-th part of it (y=1,2, k) and the other coordinates are 0. E.g., x y = (0,x,0,0) R kn (here k=4, y=2) Now we can understand the n-dimensional decision rule: Predict y iff 8 y 2 {1, k}, y :=y (w yt w y T ) x 0 (**) Equivalently, in the nk-dimensional space. Predict y iff 8 y 2 {1, k}, y :=y w T (x y x y ) w T x yy 0 Conclusion: The set (x yy, + ) (x y x y, +) is linearly separable from the set (-x yy, - ) using the linear separator w 2 R kn We solved the voroni diagram challenge. 34

30 Constraint Classification Training: [We first explain via Kesler s construction; then show we don t need it] Given a data set {(x,y)}, (m examples) with x 2 R n, y 2 {1,2, k} create a binary classification task (in R kn ): (x y - x y, +), (x y x y -), for all y : = y (2m(k-1) examples) Here x y 2 R kn Use your favorite linear learning algorithm to train a binary classifier. Prediction: Given an nk dimensional weight vector w and a new example x, predict: argmax y w T x y 35

31 Transform Examples 2>1 2>3 2>4 Details: Kesler Construction & Multi-Class Separability If (x,i) was a given n- dimensional example (that is, x has 2>4 is labeled 2>1 i, then x ij, 8 j=1, k, j:= i, are positive examples in the nk-dimensional space. 2>3 x ij are negative examples. i>j f i (x) - f j (x) > 0 w i x - w j x > 0 W X i - W X j > 0 W (X i - X j ) > 0 W X ij > 0 X i = (0,x,0,0) R kd X j = (0,0,0,x) R kd X ij = X i - X j = (0,x,0,-x) W = (w 1,w 2,w 3,w 4 ) R kd 36

32 Kesler s Construction (1) y = argmax i=(r,b,g,y) w i.x w i, x R n Find w r,w b,w g,w y R n such that w r.x > w b.x w r.x > w g.x w r.x > w y.x H = R kn 37

33 Kesler s Construction (2) Let w = (w r,w b,w g,w y ) R kn Let 0 n, be the n-dim zero vector x -x -x x w r.x > w b.x w.(x,-x,0 n,0 n ) > 0 w.(-x,x,0 n,0 n ) < 0 w r.x > w g.x w.(x,0 n,-x,0 n ) > 0 w.(-x,0 n,x,0 n ) < 0 w r.x > w y.x w.(x,0 n,0 n,-x) > 0 w.(-x,0 n,0 n,x) < 0 38

34 Kesler s Construction (3) Let w = (w 1,..., w k ) R n x... x R n = R kn x ij = (0 (i-1)n, x, 0 (k-i)n ) (0 (j-1)n, x, 0 (k-j)n ) R kn x Given (x, y) R n x {1,...,k} For all j y (all other labels) Add to P + (x,y), (x yj, 1) Add to P - (x,y), ( x yj, -1) -x P + (x,y) has k-1 positive examples ( R kn ) P - (x,y) has k-1 negative examples ( R kn ) 39

35 Learning via Kesler s Construction Given (x 1, y 1 ),..., (x N, y N ) R n x {1,...,k} Create P + = P + (x i,y i ) P = P (x i,y i ) Find w = (w 1,..., w k ) R kn, such that w.x separates P + from P One can use any algorithm in this space: Perceptron, Winnow, SVM, etc. To understand how to update the weight vector in the n-dimensional space, we note that w T x yy 0 (in the nk-dimensional space) is equivalent to: (w yt w y T ) x 0 (in the n-dimensional space) MultiClass CS446 Spring 17 40

36 Perceptron in Kesler Construction A perceptron update rule applied in the nk-dimensional space due to a mistake in w T x ij 0 Or, equivalently to (w it w jt ) x 0 (in the n-dimensional space) Implies the following update: Given example (x,i) (example x 2 R n, labeled i) 8 (i,j), i,j = 1, k, i := j (***) If (w it - w jt ) x < 0 (mistaken prediction; equivalent to w T x ij < 0 ) w i w i +x (promotion) and w j w j x (demotion) Note that this is a generalization of balanced Winnow rule. Note that we promote w i and demote k-1 weight vectors w j 41

37 Conservative update The general scheme suggests: Given example (x,i) (example x 2 R n, labeled i) 8 (i,j), i,j = 1, k, i := j (***) If (w it - w jt ) x < 0 (mistaken prediction; equivalent to w T x ij < 0 ) w i w i +x (promotion) and w j w j x (demotion) Promote w i and demote k-1 weight vectors w j A conservative update: (SNoW and LBJava s implementation): In case of a mistake: only the weights corresponding to the target node i and that closest node j are updated. Let: j* = argmax j=1, k w jt x (highest activation among competing labels) If (w it w T j* ) x < 0 (mistaken prediction) w i w i +x (promotion) and w j* w j* x (demotion) Other weight vectors are not being updated. 42

38 Multiclass Classification Summary 1: Multiclass Classification From the full dataset, construct three binary classifiers, one for each class w bluet x > 0 for blue inputs w orgt x > 0 for orange inputs w blackt x > 0 for black inputs Notation: Score for blue label Winner Take All will predict the right answer. Only the correct label will have a positive score 43

39 Multiclass Classification Summary 2: One-vs-all may not always work Red points are not separable with a single binary classifier The decomposition is not expressive enough! w bluet x > 0 for blue inputs w orgt x > 0 for orange inputs w blackt x > 0 for black inputs??? 44

40 Easy to learn Summary 3: Local Learning: One-vs-all classification Use any binary classifier learning algorithm Potential Problems Calibration issues We are comparing scores produced by K classifiers trained independently. No reason for the scores to be in the same numerical range! Train vs. Train Does not account for how the final predictor will be used Does not optimize any global measure of correctness Yet, works fairly well In most cases, especially in high dimensional problems (everything is already linearly separable). 45

41 Summary 4: Global Multiclass Approach [Constraint Classification, Har-Peled et. al 02] Create K classifiers w 1, w 2,, w K. ; Predict with WTA: argmax i w it x But, train differently: For examples with label i, we want w it x > w jt x for all j Training: For each training example (x i, y i ) : y arg max j w j T φ(x i, y i ) if y y i w yi w yi + ηx i w y w y ηx i (promote) (demote) η: learning rate 46

42 Significance The hypothesis learned above is more expressive than when the OvA assumption is used. Any linear learning algorithm can be used, and algorithmic-specific properties are maintained (e.g., attribute efficiency if using winnow.) E.g., the multiclass support vector machine can be implemented by learning a hyperplane to separate P(S) with maximal margin. As a byproduct of the linear separability observation, we get a natural notion of a margin in the multi-class case, inherited from the binary separability in the nk-dimensional space. Given example x ij 2 R nk, margin(x ij,w) = min ij w T x ij Consequently, given x 2 R n, labeled i margin(x,w) = min j (w it - w jt ) x 47

43 Margin MultiClass CS446 Spring 17 48

44 Multiclass Margin MultiClass CS446 Spring 17 49

45 Constraint Classification The scheme presented can be generalized to provide a uniform view for multiple types of problems: multi-class, multi-label, categoryranking Reduces learning to a single binary learning task Captures theoretical properties of binary algorithm Experimentally verified Naturally extends Perceptron, SVM, etc... It is called constraint classification since it does it all by representing labels as a set of constraints or preferences among output labels. 50

46 Multi-category to Constraint Classification The unified formulation is clear from the following examples: Multiclass Just like in the multiclass we learn one w (x, A) (x, (A>B, A>C, A>D) ) i 2 R n for each label, the same is done for Multilabel multi-label and ranking. The (x, (A, B)) (x, ( (A>C, A>D, B>C, B>D) ) weight vectors are updated according with the Label Ranking requirements from y 2 S k (x, (5>4>3>2>1)) (x, ( (5>4, 4>3, 3>2, 2>1) ) In all cases, we have examples (x,y) with y S k Where S k : partial order over class labels {1,...,k} defines preference relation ( > ) for class labeling Consequently, the Constraint Classifier is: h: X S k h(x) is a partial order h(x) is consistent with y if (i<j) y (i<j) h(x) (Consult the Perceptron in Kesler construction slide) 51

47 Properties of Construction (Zimak et. al 2002, 2003) Can learn any argmax v i.x function (even when i isn t linearly separable from the union of the others) Can use any algorithm to find linear separation Perceptron Algorithm ultraconservative online algorithm [Crammer, Singer 2001] Winnow Algorithm multiclass winnow [ Masterharm 2000 ] Defines a multiclass margin by binary margin in R kd multiclass SVM [Crammer, Singer 2001] 52

48 Margin Generalization Bounds Linear Hypothesis space: h(x) = argsort v i.x v i, x R d argsort returns permutation of {1,...,k} CC margin-based bound = min (x,y)s min (i < j)y v i.x v j.x err D (h) C R 2 m ln() 2 m - number of examples R - max x x - confidence C - average # constraints 53

49 VC-style Generalization Bounds Linear Hypothesis space: h(x) = argsort v i.x v i, x R d argsort returns permutation of {1,...,k} CC VC-based bound err D (h) err(s,h) m - number of examples d - dimension of input space delta - confidence k - number of classes kdlog(mk /d) ln m Performance: even though this is the right thing to do, and differences can be observed in low dimensional cases, in high dimensional cases, the impact is not always significant. 54

50 Beyond MultiClass Classification Ranking category ranking (over classes) ordinal regression (over examples) Multilabel x is both red and blue Complex relationships x is more red than blue, but not green Millions of classes sequence labeling (e.g. POS tagging) The same algorithms can be applied to these problems, namely, to Structured Prediction This observation is the starting point for CS546. MultiClass CS446 Spring 17 55

51 (more) Multi-Categorical Output Tasks Sequential Prediction (y {1,...,K} + ) e.g. POS tagging ( (NVNNA) ) This is a sentence. D V D N e.g. phrase identification Many labels: K L for length L sentence Structured Output Prediction (y C({1,...,K} + )) e.g. parse tree, multi-level phrase identification e.g. sequential prediction Constrained by domain, problem, data, background knowledge, etc... MultiClass CS446 Spring 17 56

52 Semantic Role Labeling: A Structured-Output Problem For each verb in a sentence 1. Identify all constituents that fill a semantic role 2. Determine their roles Core Arguments, e.g., Agent, Patient or Instrument Their adjuncts, e.g., Locative, Temporal or Manner A0 : leaver A2 : benefactor I left my pearls to my daughter-in-law in my will. A1 : thing left AM-LOC MultiClass CS446 Spring 17 57

53 Semantic Role Labeling Just like in the multiclass case we can think about local vs. global predictions. Local: each component learned separately, w/o thinking about other components. Global: learn to predicting the whole structure. Algorithm: essentially the same as CC A0 - A1 A2 AM-LOC I left my pearls to my daughter-in-law in my will. Many possible valid outputs Many possible invalid outputs Typically, one correct output (per input) MultiClass CS446 Spring 17 58